The Split Attractor Flow Conjecture (SAFC),
originally proposed in [1]}, [2]} and sharpened in [3]},
posits that for any \(\gamma \in \Gamma \)  and \(z\in \Pi \) , the BPS index \(\Omega _Z(\gamma )\)  can be computed by summing over
a finite number of such attractor flow trees, weighted by the
product of attractor indices \(\Omega _\star (\gamma _i)\)  and by some combinatorial factor
obtained by applying the wall-crossing formula at each vertex.The original
formulation of the conjecture
relied on the primitive wall-crossing formula and
overlooked issues arising when some of the constituents
carry non-primitive or identical charges. In §REF ,
using insights from [3]}, [5]}, [6]}, [7]}
we give a more precise
version of the conjecture in terms of the rational DT invariants
\(\bar{\Omega }(\gamma )\)  defined in (REF ).
If correct, this picture
provides a categorization (which we like to call dendroscopy)
of the BPS spectrum at \(z\in \Pi \)  into different types, each having
a simple region of stability delimited by the first splitting
at the root of the tree,
and reduces the determination of the BPS spectrum to the computation of the attractor invariants
\(\Omega _\star (\gamma _i)\) . Unfortunately, for a compact CY threefold \(\mathfrak {Y}\) , the computation of these invariants seems very difficult and the Split Attractor Flow Conjecture is still wide open,
despite some encouraging results [3]}, [9]}, [10]}, [5]}, [12]}, [13]}, [14]}.
